Let the Goal Suggest the Next Step
In How to Read a Mathematical Proof, we tracked the path from a theorem’s hypotheses to its conclusion. Before writing that path, it can help to inspect the conclusion and ask what would be enough to establish it. This is working backward: begin with the goal, use its form to identify a more manageable target, and continue until the target connects to the hypotheses or to known facts.
For example, a conclusion about an integer being divisible by a number asks for a particular factorization. A conclusion that a product is positive asks us to establish suitable signs for its factors. An inequality involving a polynomial may become easier to understand after factoring or completing a square. In each case, the conclusion tells us what kind of evidence the proof needs.
This is a discovery method. It helps locate a route, but a proof must still justify every step from the assumptions to the conclusion. If a backward analysis has the form “the desired conclusion would follow if we had \(R\), and \(R\) would follow from the hypothesis,” the written proof should present the supported implications in the forward order: $$ \text{hypothesis}\Longrightarrow R\Longrightarrow\text{conclusion}. $$
Unpack the Conclusion Before Calculating
A useful first step is to translate the conclusion into its definition or into a familiar equivalent statement. Suppose the goal is to prove that an integer \(N\) is divisible by a nonzero integer \(m\). By the definition of divisibility, the endpoint is to find an integer \(r\) such that \(N=mr\). This translation clarifies what an algebraic calculation must produce. It also identifies a condition that cannot be omitted: the proposed quotient \(r\) must be an integer.
For inequalities, the same habit can reveal useful factors. If the goal is \(A>0\) and \(A\) can be written as a product, one possible plan is to establish that both factors are positive, or that both are negative. Which plan works depends on the hypotheses. If the goal is \(A\geq0\), a square may be useful because the square of a real number is nonnegative.
The aim is not to guess a sequence of algebraic tricks. It is to identify a condition that is both sufficient for the conclusion and reachable from the assumptions. The domain matters: a rearrangement that is valid for every real number can be used without an additional sign condition, while division by an expression requires knowing that the expression is nonzero.
Keep the conclusion in view, including its quantifiers, domain, and any strict or non-strict inequality.
Use its definition or an equivalent expression to see what the final line must establish.
Look for a simpler condition that implies the goal and that might follow from the hypotheses.
Confirm whether a step is reversible and record any needed conditions, such as a sign or a nonzero denominator.
Start from the hypotheses, establish the intermediate facts, and finish by matching the stated conclusion.
Use the Goal to Find a Useful Intermediate Statement
Consider a claim whose conclusion is a product inequality. Working backward, we can inspect the factors and ask what their signs would need to be. That inspection does not itself prove the claim: the signs must be derived from the hypothesis. Once they are, the proof proceeds forward by multiplying positive quantities.
Worked Example: Turning a Product Goal into Sign Conditions
Theorem. If \(x>-1\), then \((x+1)(x+3)>0\).
Backward analysis. The conclusion is that a product is positive. One sufficient condition is that both factors are positive. The hypothesis \(x>-1\) gives \(x+1>0\) immediately. Adding \(3\) to both sides of \(x>-1\) gives \(x+3>2>0\). Thus the hypothesis supplies both sign conditions that the goal suggests.
Proof. Let \(x\in\mathbb R\) and suppose \(x>-1\). Adding \(1\) gives \(x+1>0\). Adding \(3\) gives \(x+3>2>0\). Since the product of two positive real numbers is positive, $$ (x+1)(x+3)>0. $$ This is the stated conclusion, so the theorem is proved.
The backward analysis identified a route, while the proof states the reasoning from the hypothesis to the result. Notice also that proving only \(x+1>0\) would not be enough: the sign of the other factor is needed for this particular argument. The hypothesis provides it, but it must still be checked.
A target can often be rewritten in more than one way. For instance, an inequality may be transformed by adding the same quantity to both sides, or a polynomial may be factored. When a transformation is reversible, the original and transformed statements are equivalent for the inputs under consideration. When it is not reversible, it may give a useful sufficient condition but cannot automatically replace the goal.
Worked Example: Rewriting a Universal Inequality as a Square
Theorem. For every real number \(x\), \(x^2+1\geq 2x\).
Backward analysis. Move the right-hand side to the left. The desired inequality is equivalent to $$ x^2-2x+1\geq0. $$ The left side is a perfect square, so it is enough to recognize it as \((x-1)^2\). The square of a real number is nonnegative for every real input. This suggests the proof.
Proof. Let \(x\in\mathbb R\). Since a square of a real number is nonnegative, $$ (x-1)^2\geq0. $$ Expanding gives \(x^2-2x+1\geq0\). Adding \(2x\) to both sides yields \(x^2+1\geq2x\). The claim holds for every real \(x\).
The expansion and rearrangement preserve the inequality because the same quantity is added to both sides. There is no division by an expression whose sign or nonzero status would need to be checked. The equality case is also visible: the square equals zero exactly when \(x-1=0\), so equality holds at \(x=1\). Equality is not needed to prove the inequality, but noticing it can help check whether the result is plausible.
Equivalent Steps and One-Way Steps
When working backward, label the kind of connection you are using. An equivalence allows movement in both logical directions. For example, adding the same real number to both sides of an equation preserves equivalence. A one-way implication gives only a sufficient route: if \(R\) implies the goal, then proving \(R\) is enough, but the goal need not imply \(R\).
This distinction is especially important with inequalities. Multiplication by a positive number preserves the direction of a strict inequality; multiplication by a negative number reverses it. If the sign of the multiplier is unknown, multiplying an inequality by it does not give one fixed direction without considering cases. Likewise, multiplying by \(x\) and then dividing by \(x\) is reversible only when \(x\ne0\), and the direction of an inequality after division depends on the sign of \(x\).
| Backward move | What to verify | How to use it in a proof |
|---|---|---|
| Expand or factor an expression | Do the expressions remain equal? | State the identity, then use the form suited to the goal. |
| Add or subtract the same quantity | Is the same quantity applied to both sides? | Carry out the operation while preserving the relation. |
| Multiply or divide an inequality | Is the multiplier positive, negative, or zero? | Establish its sign first, and reverse the inequality when required. |
| Divide an equation by an expression | Has that expression been shown to be nonzero? | State the nonzero condition before dividing. |
| Replace the goal by a sufficient condition | Does the condition actually imply the goal? | Prove the sufficient condition from the hypotheses, then state the implication. |
The direction of an implication is not a matter of presentation. If the analysis says that \(R\) is sufficient for \(S\), then \(R\Longrightarrow S\). Establishing \(S\) does not in general establish \(R\). A sound proof plan must move from known facts toward the goal, not assume the goal in order to obtain something that would imply it.
Worked Example: Finding the Required Integer in a Divisibility Claim
Theorem. If an integer \(n\) is divisible by \(3\), then \(n^2\) is divisible by \(9\).
Backward analysis. By the definition of divisibility by \(9\), the conclusion requires an integer \(r\) such that \(n^2=9r\). The hypothesis that \(n\) is divisible by \(3\) gives a representation \(n=3k\) for some integer \(k\). Substitution suggests $$ n^2=(3k)^2=9k^2. $$ The remaining check is that \(k^2\) is an integer.
Proof. Let \(n\in\mathbb Z\) be divisible by \(3\). By the definition of divisibility, there is an integer \(k\) such that \(n=3k\). Therefore $$ n^2=(3k)^2=9k^2. $$ Because \(k\) is an integer, \(k^2\) is an integer. Taking \(r=k^2\), we have expressed \(n^2\) as \(9r\) with \(r\in\mathbb Z\). By the definition of divisibility, \(n^2\) is divisible by \(9\).
Here the endpoint was not merely the equation \(n^2=9k^2\). The definition requires a factor multiplied by \(9\) that is an integer, so the proof explicitly identifies \(r\) and verifies its domain. That final check completes the argument.
Do Not Assume the Conclusion During the Analysis
Working backward can feel as if one is starting from the conclusion and manipulating it. There is a danger here: if a line in the analysis depends on the conclusion itself, the reasoning may be circular. The analysis is useful for discovering what would suffice, but it cannot treat the result to be proved as an available fact.
For example, suppose a goal is \(A>0\) and the analysis says, “If \(A>0\), then \(A^2>0\).” This observation may be true under suitable conditions, but it does not help establish \(A>0\) from the original hypotheses. It starts by assuming the very statement that is the goal. Instead, ask whether the hypotheses establish another fact that implies \(A>0\), such as a positive factorization or a comparison with zero.
A second danger is accepting a one-way transformation in the wrong direction. If \(S\) implies \(R\), then showing \(R\) does not prove \(S\). In ordinary language, “being a square greater than zero implies being positive” does not mean that every positive number has already been shown to be a square greater than zero. The direction needed by the proof must match the direction established by the analysis.
From a Backward Plan to a Forward Proof
A backward plan is often brief: “The goal would follow from \(R\); the hypothesis gives \(R\).” A readable proof makes the relationship explicit and supplies the needed details. It begins by selecting arbitrary objects in the stated domain, introduces the hypotheses, derives intermediate claims, and finishes by matching the original conclusion. The method in Introduction to Mathematical Proof gives the general structure for proving a universal conditional statement.
Before writing, it can help to make a small two-column record. On one side, list what the conclusion requires. On the other, list what the hypotheses give. Look for a justified bridge between the two. In the divisibility example, the conclusion required an integer factor after \(9\), while the hypothesis supplied an integer factor after \(3\). Substitution connected those forms. In the square inequality, the conclusion became a nonnegative square, which was available for every real number.
The bridge need not be a calculation. It may be a definition, an earlier result, a case division, or the construction of an object. Whatever the route, test its conditions before using it. If a familiar theorem supplies the needed bridge, cite it by name and verify that its hypotheses apply rather than proving it again.
There is no requirement that a finished proof mention the backward analysis. Include it only if it helps the reader understand why the proof takes its chosen route. The final proof should stand on its own: its starting facts are clear, its inferences are valid, and its endpoint is the stated conclusion.
Check Your Understanding
Use the distinction between proof planning and proof writing to respond to each item.
- For a goal that an integer \(N\) is divisible by \(7\), what precise form does the definition tell you to seek?
- In the proof that \(x>-1\) implies \((x+1)(x+3)>0\), why must the proof check both factors?
- For \(x^2+1\geq2x\), what equivalent expression reveals the useful nonnegative quantity?
- Why does the divisibility proof need to say that \(k^2\) is an integer, rather than stopping after \(n^2=9k^2\)?
- If working backward shows that the conclusion would follow from \(R\), what must the forward proof establish before using that observation?
- When is division by an expression valid in a proof, and what additional issue arises if the proof involves an inequality?